Robust Statistics and Outliers

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<span>
<a href="https://psychmethods.github.io/coursenotes/" target="_blank">Methods in Psychological Research</a>
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# Robust Statistics

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## What is an Outlier?

1. Something that is situated away from or classed differently from a main or related body

2. A statistical observation that is markedly different in value from the others of the sample  
]

.pull-right[  
*"An important kind of deviation is an outlier, an individual value that falls outside the overall pattern."*  
]

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## Robustness

- Robust statistics are less sensitive to outliers
- Most common statistics 
  - (means, standard deviations, correlations) 
  - are highly sensitive to outliers

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## Dealing with Outliers

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## Example: Math Test Scores

Consider Tukey's scores for the last 5 math tests: 88, 90, 55, 94, and 89.

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## Calculations: Mean

``` r
scores <- c(88, 90, 55, 94, 89)  
mean(scores)  
```

```
## [1] 83.2
```

55+88+89+90+94 = 416

416 / 5 = 83.2

The mean is 83.2

]
.pull-right[  
**Without Outlier:**

``` r
nooutlier <- c(88, 90, 94, 89)  
mean(nooutlier)  
```

```
## [1] 90.25
```

88+89+90+94 = 361

361 / 4 = 90.25

The mean is 90.25

]

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## Calculations: Median

``` r
median(scores)  
```

```
## [1] 89
```

Median: 55,88,_89_,90,94

The median is 89

]

``` r
median(nooutlier)  
```

```
## [1] 89.5
```

Median: 88,_89,90_,94
(89+90) / 2
The median is 89.5

]

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## Handling Outliers

- If you remove outliers:
  - Document it
  - Explain why (e.g., obvious error, impossible number)

- If you keep outliers:
  - Be aware they might skew your results
  - Consider using robust statistics like median or trimmed mean

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## 5-Number Summary

A robust method for summarizing distributions:

1. Minimum
2. First Quartile (Q1)
3. Median
4. Third Quartile (Q3)
5. Maximum

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## Boxplot

![](path/to/boxplot_diagram.png)  
Graphical representation of the 5-number summary

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## Diagnosing Outliers

Common methods for identifying outliers:

1. **IQR Rule**:  
   - Lower fence: Q1 - 1.5 * IQR  
   - Upper fence: Q3 + 1.5 * IQR

2. **Standard Deviation Rule**:  
   - Values beyond 2.5 standard deviations from the mean

3. **More complex algorithms**:  
   - Specific robust methods like M-estimators

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## Beeswarm Boxplot

A variant of the boxplot that shows individual data points:

``` r
if(!require(beeswarm)) install.packages("beeswarm")  
data(breast)  
beeswarm(time_survival ~ event_survival, data = breast, method = 'swarm', pch = 16,  
         pwcol = as.numeric(ER), xlab = '',  
         ylab = 'Follow-up time (months)',  
         labels = c('Censored', 'Metastasis'))  
boxplot(time_survival ~ event_survival, data = breast, add = TRUE,  
        names = c("", ""), col="#0000ff22")  
```

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# Wrapping Up

- The decision to remove or keep outliers depends on your research context  
- Always be transparent about your choices